Rozansky-Witten theory is a physical 3d TQFT which depends on a hyperkahler manifold. Roberts and Willerton studying topics related to the partition function of this theory highlighted the relevance of the derived category of coherent sheaves on the hyperkahler manifold. In this talk, I will discuss the Hopf algebra object in the derived category of coherent sheaves motivated by a rigorous construction of an extended TQFT for Rozansky-Witten theory.

In this talk I shall describe an ongoing joint project with Festuccia, Winding and Zabzine, aiming at generalizing the Donaldson-Witten theory in a natural way. The DW theory computes integrals of certain characteristic classes over the moduli space of anti-self-dual Yang-Mills instantons. We attempt to generalize it in two directions. First, the instanton configuration $F^+=0$ (where $F$ is the Yang-Mills field strength) is generalized to $PF=0$, where $P$ is a projector projecting $F$ to a rank 3 sub-bundle of 2-forms. Second, we add equivariance.

The idea of topologically twisting a supersymmetric field theory was introduced in the physics literature in order to generate interesting new examples of topological field theories. The idea is very general, but systematically realising the examples it produces using mathematical models for topological quantum field theory (such as the functorial axioms of Atiyah-Segal or the theory of $E_n$-algebras) is not always possible.

I will describe examples of holomorphic N=1 super-symmetric vertex algebras with small (non-zero) values of the elliptic genus. I will speculate on a relation to certain patterns in the theory of topological modular forms.

The goal of my talk will be to discuss the relation between two approaches to the geometric Langlands program. The first has been proposed by Beilinson and Drinfeld, using ideas and methods from conformal field theory (CFT). The second was initiated by Kapustin and Witten based on a topological version of four-dimensional maximally supersymmetric Yang--Mills theory and its reduction to a two-dimensional topological sigma model.

This is a join work with T. Pantev. In this talk, we will discuss moduli of flat bundles on smooth algebraic varieties, with possibly irregular singularities at infinity. For this, we use the notion of ''formal boundary'', previously studied by Ben Bassat-Temkin, Efimov and Hennion-Porta-Vezzosi, as well as the moduli of flat bundles at infinity. We prove that the fibers of the restriction map to infinity are representable. We also prove that this restriction map has a canonical Lagrangian structure in the sense of shifted symplectic geometry.

The based loop group is an infinite-dimensional manifold equipped with a Hamiltonian action of a finite dimensional torus. This was studied by Atiyah and Pressley. We investigate the Duistermaat--Heckman distribution using the theory of hyperfunctions. In applications involving Hamiltonian actions on infinite-dimensional manifolds, this theory is necessary to accommodate the existence of the infinite order differential operators which aries from the isotropy representation on the tangent spaces to fixed points. (Joint work with James Mracek)

I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to $\mathbb{E}_n$-algebras.I will also give the main ingredients of the proof of Kontsevich's conjecture, which is a joint work in progress with Damien Lejay.

I will review the construction of ''higher operations'' on local and extended operators in topological field theory, and some applications of this construction in supersymmetric field theory. In particular, the higher operation on supersymmetric local operators in a 3d N=4 theory turns out to be induced by the holomorphic Poisson structure on the moduli space of the theory.